Pre-Algebra Examples

$14 x^{2} + 33 \cdot 4 = 15 x + 3$

Step 1

Multiply $33$ by $4$ .

$14 x^{2} + 132 = 15 x + 3$

Step 2

Subtract $15 x$ from both sides of the equation.

$14 x^{2} + 132 - 15 x = 3$

Step 3

Move all terms to the left side of the equation and simplify.

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Step 3.1

Subtract $3$ from both sides of the equation.

$14 x^{2} + 132 - 15 x - 3 = 0$

Step 3.2

Subtract $3$ from $132$ .

$14 x^{2} - 15 x + 129 = 0$

Step 4

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 5

Substitute the values $a = 14$ , $b = - 15$ , and $c = 129$ into the quadratic formula and solve for $x$ .

$\frac{15 \pm \sqrt{{(- 15)}^{2} - 4 \cdot (14 \cdot 129)}}{2 \cdot 14}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $- 15$ to the power of $2$ .

$x = \frac{15 \pm \sqrt{225 - 4 \cdot 14 \cdot 129}}{2 \cdot 14}$

Step 6.1.2

Multiply $- 4 \cdot 14 \cdot 129$ .

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Step 6.1.2.1

Multiply $- 4$ by $14$ .

$x = \frac{15 \pm \sqrt{225 - 56 \cdot 129}}{2 \cdot 14}$

Step 6.1.2.2

Multiply $- 56$ by $129$ .

$x = \frac{15 \pm \sqrt{225 - 7224}}{2 \cdot 14}$

Step 6.1.3

Subtract $7224$ from $225$ .

$x = \frac{15 \pm \sqrt{- 6999}}{2 \cdot 14}$

Step 6.1.4

Rewrite $- 6999$ as $- 1 (6999)$ .

$x = \frac{15 \pm \sqrt{- 1 \cdot 6999}}{2 \cdot 14}$

Step 6.1.5

Rewrite $\sqrt{- 1 (6999)}$ as $\sqrt{- 1} \cdot \sqrt{6999}$ .

$x = \frac{15 \pm \sqrt{- 1} \cdot \sqrt{6999}}{2 \cdot 14}$

Step 6.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{15 \pm i \sqrt{6999}}{2 \cdot 14}$

Step 6.2

Multiply $2$ by $14$ .

$x = \frac{15 \pm i \sqrt{6999}}{28}$

Step 7

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $- 15$ to the power of $2$ .

$x = \frac{15 \pm \sqrt{225 - 4 \cdot 14 \cdot 129}}{2 \cdot 14}$

Step 7.1.2

Multiply $- 4 \cdot 14 \cdot 129$ .

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Step 7.1.2.1

Multiply $- 4$ by $14$ .

$x = \frac{15 \pm \sqrt{225 - 56 \cdot 129}}{2 \cdot 14}$

Step 7.1.2.2

Multiply $- 56$ by $129$ .

$x = \frac{15 \pm \sqrt{225 - 7224}}{2 \cdot 14}$

Step 7.1.3

Subtract $7224$ from $225$ .

$x = \frac{15 \pm \sqrt{- 6999}}{2 \cdot 14}$

Step 7.1.4

Rewrite $- 6999$ as $- 1 (6999)$ .

$x = \frac{15 \pm \sqrt{- 1 \cdot 6999}}{2 \cdot 14}$

Step 7.1.5

Rewrite $\sqrt{- 1 (6999)}$ as $\sqrt{- 1} \cdot \sqrt{6999}$ .

$x = \frac{15 \pm \sqrt{- 1} \cdot \sqrt{6999}}{2 \cdot 14}$

Step 7.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{15 \pm i \sqrt{6999}}{2 \cdot 14}$

Step 7.2

Multiply $2$ by $14$ .

$x = \frac{15 \pm i \sqrt{6999}}{28}$

Step 7.3

Change the $\pm$ to $+$ .

$x = \frac{15 + i \sqrt{6999}}{28}$

Step 8

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $- 15$ to the power of $2$ .

$x = \frac{15 \pm \sqrt{225 - 4 \cdot 14 \cdot 129}}{2 \cdot 14}$

Step 8.1.2

Multiply $- 4 \cdot 14 \cdot 129$ .

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Step 8.1.2.1

Multiply $- 4$ by $14$ .

$x = \frac{15 \pm \sqrt{225 - 56 \cdot 129}}{2 \cdot 14}$

Step 8.1.2.2

Multiply $- 56$ by $129$ .

$x = \frac{15 \pm \sqrt{225 - 7224}}{2 \cdot 14}$

Step 8.1.3

Subtract $7224$ from $225$ .

$x = \frac{15 \pm \sqrt{- 6999}}{2 \cdot 14}$

Step 8.1.4

Rewrite $- 6999$ as $- 1 (6999)$ .

$x = \frac{15 \pm \sqrt{- 1 \cdot 6999}}{2 \cdot 14}$

Step 8.1.5

Rewrite $\sqrt{- 1 (6999)}$ as $\sqrt{- 1} \cdot \sqrt{6999}$ .

$x = \frac{15 \pm \sqrt{- 1} \cdot \sqrt{6999}}{2 \cdot 14}$

Step 8.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{15 \pm i \sqrt{6999}}{2 \cdot 14}$

Step 8.2

Multiply $2$ by $14$ .

$x = \frac{15 \pm i \sqrt{6999}}{28}$

Step 8.3

Change the $\pm$ to $-$ .

$x = \frac{15 - i \sqrt{6999}}{28}$

Step 9

The final answer is the combination of both solutions.

$x = \frac{15 + i \sqrt{6999}}{28}, \frac{15 - i \sqrt{6999}}{28}$